Question

The tangent of the graph of the function y = f(x) at the point with abscissa x = 1 form an angle of $\frac{\pi }{6}$ and at the point x = 2 an angle of $\frac{\pi }{3}$ and at the point x = 3 angle of
​$\frac{\pi }{4}$. The value of ${\int }_1^3{f}^{\prime }\left(x\right)f"\left(x\right)dx+{\int }_1^{3}f"\left(x\right)dx(f"(x\left)\right)$ is suppose to be continuous) is

• A. $\frac{4\sqrt{3}-1}{3\sqrt{3}}$
• B. $\frac{3\sqrt{3}-1}{2}$
• C. $\frac{4-\sqrt{3}}{3}$
• D. $Noneofthese$

Answer 1

The correct option is B $\frac{3\sqrt{3}-1}{2}$

$f^{\prime }\left(x\right)=n(1+x{)}^{n-1},f"(x)=n(n-1\left)\right(1+x{)}^{n-2}{f}^n\left(x\right)=n!,fn\left(0\right)=n!\Rightarrow 1+n+\frac{n(n-1)}{2!}+....+\frac{n!}{n!}{=}^n{C}_0{+}^n{C}_1{+}^n{C}_2+...{+}^n{C}_n=2^n$